Lagrangian and Hamiltonian mechanics

Constraints

Imposing constraints on a system is simply another way of stating that there are forces present in the problem that cannot be specified directly, but are known in term of their effect on the motion of the system.  Holonomic constraints are constraints of the form
f_m(r₁,r₂,r₃,...,r_n,t) = 0,   m = 1, 2, 3, ... , k.
They reduce the number of degrees of freedom of the system;  k equations of constraints reduce the number of the degree of freedom of an n-particle system from 3n to 3n - k, if the constraints are holonomic.
If the constraints are holonomic, then the forces of constraints do no virtual work.
Consider a virtual displacement of the system, i.e. an infinitesimal change in the coordinates of the system, denoted by dr_i, consistent with the constraints imposed on the system at a given instant t.  The work done by the force in the virtual displacement dr_i is called the virtual work.  For holonomic constraints, the forces of constraints are perpendicular to the virtual displacements and do no virtual work.

Generalized coordinates

Any set of independent quantities q₁, q₂, ... , q_s, which completely define the position of the system with s degrees of freedom, are called generalized coordinates of the system, and the derivatives  are called generalized velocities.
Examples:

• A particle is constraint to move in the x-y plane, the equation of constraint is z = 0, the constraint is holonomic.  Possible generalized coordinates for the system with two degrees of freedom are x, y;  r, φ;  ... .
• A particle is constraint to move on a circle in the x-y plane, the equations of constraints are z = 0,  x²  + y² - r²  = 0.  The constraints are holonomic.  Possible generalized coordinates for the system with 1 degree of freedom are φ ; φ³, ... .

The generalized coordinates q₁, ... , q_s can be expressed in terms of the Cartesian coordinates the system.
q₁ = q₁(r_1, r_{2, ... ,} r_n ), ... , q_n-m(r_1, r_{2, ... ,} r_n ).
These equations, together with the equation of constraints, can be inverted to find the r's in terms of the q's.

Lagrangian Mechanics

Assume a system has n independent generalized coordinates {q_i}.  Assume that the generalized applied forces
{Q_j} = {∑_iF_i·∂r_i/∂q_j}
are given by
Q_j = -∂U/∂q_j  or  -∂U/∂q_j + d/dt(∂U/∂(dq_j/dt)),
with U some scalar function, i.e. the generalized applied forces are derivable from a potential.  Then the equations of motion may be obtained from Lagrange's equations,

d/dt(∂L/∂(dq_i/dt)) -  ∂L/∂q_i = 0,

where L = T - U is the Lagrangian of the system.  L is a function of the coordinates q_i and the velocities dq_i/dt.

Example of a velocity dependent potential

If not all the forces acting on the system are derivable from a potential, then Lagrange's equations can be written in the form

d/dt(∂L/∂(dq_i/dt)) -  ∂L/∂q_i = Q_i,

where L contains the potential of the conservative forces and Q_i represents the generalized forces not arising from a potential.

Link:  Derive Lagrange's equations from D'Alembert's Principle

Define the generalized momentum or conjugate momentum or canonical momentum through
∂L/∂(dq_i/dt) = p_i.
If the Lagrangian does not contain a given coordinate q_j then the coordinate is said to be cyclic and the corresponding conjugate momentum p_j is conserved.

The Hamiltonian H of a system is given by

H(q, p, t) = ∑_i(dq_i/dt)p_i - L.  

H is a function of the generalized coordinates and momenta of the system.  The equations of motion can be obtained from Hamilton's equations,

dq_i/dt = ∂H/∂p_i,  dp_i/dt = -∂H/∂q_i.

Assume now that the generalized forces are given by Q_j = -∂U/∂q_j.

• If the Lagrangian does not explicitly depend on time, then the Hamiltonian does not explicitly depend on time and H is a constant of motion.  [If H does explicitly depend on time, H = H(t), then H is not a constant of motion.]
• If the generalized coordinates (and the constraints) do not explicitly depend on time, then H = T + U = E is the total energy of the system.  [If the generalized coordinates (or the constraints) do explicitly depend on time, then H is not the total energy of the system.]
• So only if Lagrangian does not explicitly depend on time and the generalized coordinates and the constraints do not explicitly depend on time, then H = T + U = E and the energy is a constant of motion.

Coupled small oscillations

Let L = ½∑_ij[T_ij(dq_i/dt)(dq_j/dt) - k_ijq_iq_j]  with T_ij = T_ji,  k_ij = k_ji. 

Then solutions of the form q_j = Re(A_je^iωt) can be found.
We can find the ω² from det(k_ij - ω²T_ij) = 0. 
For a system with n degrees of freedom, n characteristic frequencies ω_α can be found. 
Some frequencies may be degenerate.
For a particular frequency ω_α we solve

∑_j[k_ij - ω_α²T_ij]A_jα = 0

to find the A_jα.  

[While the secular equation det(k_ij-ω²T_ij) = 0 can in principle always be solved, it is often simpler to find the normal modes by using physical insight and noting the symmetries of the system.]

The most general solution for each coordinate q_j is a sum of simple harmonic oscillations in all of the frequencies ω_α.

q_j = Re∑_α(C_αA_jαexp(iω_αt)).